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+\newcommand{\dif}[2]{\frac{\partial#1}{\partial#2}} + \newcommand{\enorm}[1]{\left| \! \left| \! \left|#1\right| \! \right| \! \right|} \newcommand{\norm}[1]{\lVert#1\rVert} \newcommand{\abs}[1]{\lvert#1\rvert} + + + \newcommand{\showMesh}[2][]{\begin{figure}[ht] \caption{#1} \label{#2} @@ -77,24 +83,27 @@ \section{Einleitung} \subsection{Allgemein} -In dieser Arbeit beschäftigen wir uns mit Randelementmethode für die homogene Laplace-Gleichung mit Dirichlet-Randbedingungen, -\begin{align*} - - \varDelta u &= 0 &\text{ in } \Omega& \subset \R^3\\ - u &= f &\text{ auf } \Gamma&:=\partial\Omega -\end{align*} -wobei der Laplace-Operator $\varDelta u := \partial_x^2u+\partial_y^2u$ ist und $\Omega \subset \R^3$ sei eine beschränkte Teilmenge von $\R^3$ -mit Dirichlet-Rand $\Gamma := \partial \Omega$.\\ +In dieser Arbeit beschäftigen wir uns mit der Randelementmethode für die homogene Laplace-Gleichung mit Dirichlet-Randbedingungen +\begin{align} + - \varDelta u &= 0 &\text{ in } \Omega& \subset \R^3,\\ + u &= f &\text{ auf } \Gamma&:=\partial\Omega, +\end{align} +wobei $\varDelta u := \partial_x^2u+\partial_y^2u$ den Laplace-Operator bezeichnet und $\Omega \subset \R^3$ sei eine beschränkte Teilmenge von $\R^3$ mit Dirichlet-Rand $\Gamma := \partial \Omega$.\\ -$\abs{u(x)} = O(\abs{x}^{-1}) \nonumber$ -Daraus folgt: +\noindent +Wir wissen, dass die Laplace-Gleichung erfüllt wird durch: \begin{align} V\phi &= f \end{align} -Sei nun die Fundamentallösung $G$: +% $\abs{u(x)} = O(\abs{x}^{-1}) \nonumber$ +Sei nun die $G$ Fundamentallösung, +\begin{align} + G(\bs z) := \frac 1 {4 \pi} \frac 1 {\abs{\bs z}} +\end{align} +dann können wir $\tilde V$ anschreiben durch \begin{align} -\tilde V \phi (x) &:= \int G (x-y) \phi(y) dy & x\in \R^3\backslash \Gamma +\tilde V \phi (\bs x) &:= \int G (\bs x- \bs y) \phi(\bs y) d\bs y & \bs x\in \R^3\backslash \Gamma \end{align} -Dabei ist $G(z) := \frac 1 {4 \pi} \frac 1 {\abs{z}}$. Daraus folgt nun mit $\gamma_0 = \rm{spur}$ : \begin{align} V \phi &: \gamma_0 \tilde V \phi_0 diff --git a/src/compute.m b/src/compute.m index 8a79770..3bd6105 100644 --- a/src/compute.m +++ b/src/compute.m @@ -244,7 +244,13 @@ for j = 1:times %ErgebnisWerte Speichern data(size(data,1)+1,1:length(dataS)) = dataS; typeN = int2str(type); - save (['meshSave/' out typeN(typeN~=' ') '_' int2str(size(data,1))], 'coordinates', 'elements','neigh','data', 'sites') + save (['meshSave/'... + out... + typeN(typeN~=' ')... + 't' regexprep(num2str(theta,2),'\.','')... + 'n' regexprep(num2str(theta,2),'\.','')... + '_' int2str(size(data,1))]... + , 'coordinates', 'elements','neigh','data', 'sites') %Alle Relevanten zwischenInformationen Speichern % out = '_'; -- 2.47.3